There are several additional characterizations of inner product spaces in [1]} but they use BJ orthogonality relation and additivity operation in unison.
One such, for example, is the following: a real space \(X\)  of dimension at least three is an inner product space if and only if \(x\perp z\)  and \(y\perp z\)  always imply that \((x+y)\perp z\)  (see [1]}). In a similar way, it was shown in [3]} that BJ orthogonality together with additivity (and homogeneity operations)
can characterize smooth (and rotund, that is, strictly convex) norms.
